Continuous Limits of Coupled Flows in Representation Learning

2026-04-17

AI Summary

This work addresses the unclear macroscopic convergence of discrete dynamics on continuous data manifolds and the associated parameter explosion in decentralized representation learning. The authors model the process as a coupled fast-slow stochastic dynamical system on a Riemannian manifold. By leveraging measure-theoretic limits, Itô–Poisson resolvents, and a stochastic LaSalle principle, they rigorously establish— for the first time—that discrete updates converge to an overdamped Langevin stochastic differential equation. A joint Lyapunov functional is constructed to reveal the emergent mechanism underlying automatic feature disentanglement and linear separability. The theoretical analysis demonstrates that representation weights remain uniformly bounded, align precisely with the principal eigenspace of the data measure, and that the coupled flow exhibits global dissipativity, thereby guaranteeing the spontaneous emergence of orthogonally disentangled features in the stable limit.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Responsible Web: Machine-in-the-loop, human agency and autonomy
• Web Mining and Content Analysis: Models for Web evolution

Abstract

While modern representation learning relies heavily on global error signals, decentralized algorithms driven by local interactions offer a fundamental distributed alternative. However, the macroscopic convergence properties of these discrete dynamics on continuous data manifolds remain theoretically unresolved, notoriously suffering from parameter explosion. We bridge this gap by formalizing decentralized learning as a coupled slow-fast dynamical system on Riemannian manifolds. First, using measure-theoretic limits, we prove that the discrete spatial transitions converge uniformly to an overdamped Langevin stochastic differential equation. Second, via the Itô-Poisson resolvent and a stochastic extension of LaSalle's Invariance Principle, we establish that the representation weights unconditionally avoid divergence and align strictly with the principal eigenspace of the spatial measure. Finally, we construct a joint Lyapunov functional for the fully coupled spatial-parametric flow. This proves global dissipativity and demonstrates that orthogonally disentangled, linearly separable features emerge spontaneously at the stationary limit. Our framework bridges discrete algorithms with continuous stochastic analysis, providing a formal theoretical baseline for decentralized representation learning.

Problem

• decentralized representation learning
• continuous data manifolds
• parameter explosion
• macroscopic convergence
• discrete dynamics

Innovation

• decentralized representation learning
• coupled dynamical systems
• stochastic differential equations
• Riemannian manifolds
• Lyapunov functional